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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 3762–3767
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Experimental demonstration of the optical Zeno effect by scanning tunneling optical microscopy

P. Biagioni, G. Della Valle, M. Ornigotti, M. Finazzi, L. Duò, P. Laporta, and S. Longhi  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 3762-3767 (2008)
http://dx.doi.org/10.1364/OE.16.003762


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Abstract

An experimental demonstration of a classical analogue of the quantum Zeno effect for light waves propagating in engineered arrays of tunneling-coupled optical waveguides is reported. Quantitative mapping of the flow of light, based on scanning tunneling optical microscopy, clearly demonstrates that the escape dynamics of light in an optical waveguide side-coupled to a tight-binding continuum is slowed down when projective measurements, mimicked by sequential interruptions of the decay, are performed on the system.

© 2008 Optical Society of America

1. Introduction

The quantum Zeno and anti-Zeno effects, i.e. the inhibition or acceleration of the irreversible decay of an unstable quantum state into a continuum (reservoir) induced by measurements, are commonly viewed as basic manifestations of the influence of observations on the evolution of a quantum system [1

1. B. Misra and E. C. G. Sudarshan, “The Zeno’s Paradox in Quantum Theory,” J. Math. Phys. 18, 756–763 (1977). [CrossRef]

, 2

2. W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A 41, 2295–2300 (1990). [CrossRef] [PubMed]

, 3

3. P. Knight, “Watching a Laser Hot-Pot,” Nature (London) 344, 493–494 (1990). [CrossRef]

, 4

4. H. Nakazato, M. Namiki, and S. Pascazio, “Temporal behavior of Quantum Mechanical Systems,” Int. J. Mod. Phys. B 10, 247–295 (1996). [CrossRef]

, 5

5. A.G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750–R3753 (1996). [CrossRef] [PubMed]

, 6

6. M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A 61, 022105-1-022105-5 (2000). [CrossRef]

, 7

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) 405, 546–550 (2000). [CrossRef] [PubMed]

, 8

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

, 9

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

, 10

10. P. Facchi and S. Pascazio, Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001), Vol. 42, p. 147.

]. Experiments showing interruptions of Rabi oscillations or analogous forms of nearly-reversible evolution for essentially stable few level systems have been reported as a demonstration of Zeno dynamics (see e.g. [2

2. W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A 41, 2295–2300 (1990). [CrossRef] [PubMed]

, 10

10. P. Facchi and S. Pascazio, Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001), Vol. 42, p. 147.

, 11

11. E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. 97, 260402-1-260402-4 (2006). [CrossRef]

]), rising a rather broad debate on Zeno dynamics and its connection with the theory of quantum measurements (see e.g. [10

10. P. Facchi and S. Pascazio, Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001), Vol. 42, p. 147.

, 11

11. E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. 97, 260402-1-260402-4 (2006). [CrossRef]

, 12

12. P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401-1-080401-4 (2002). [CrossRef] [PubMed]

, 13

13. K. Koshino and A. Shimizu, “Quantum Zeno Effect for exponentially decaying systems ,” Phys. Rev. Lett. 92, 030401-1-030401-4 (2004).

]). It was also argued that slow down of the decay may be attained in purely classical systems [14

14. A. Peres, “Zeno Paradox in Quantum Theory,” Am J. Phys.48, 931–932 (1980); G.S. Agarwal and S. P. Tewari, “An all-optical realization of quantum zeno effect,” Phys. Lett. A 185, 139–142 (1994); M. Kitano, “Quantum Zeno Effect and intracavity polarization filters,” Opt. Commun. 141, 39–42 (1997); V. Kidambi, A. Widom, C. Lerner, and Y. N. Srivastava, “Photon polarization measurements without the Quantum Zeno Effect,” Am. J. Phys. 68, 475–481 (2000); K. Yamane, M. Ito, and M. Kitano, “Quantum Zeno Effect in Optical Fibers,” Opt. Commun. 192, 299–307 (2001). [CrossRef]

] that can mimic the dynamics of a two-level quantum system. An experimental demonstration of the Zeno effect in a truly decaying system is very hard to be observed in microscopic quantum systems because the required measurement intervals are too short and decay acceleration (anti-Zeno effect) appears to be much more ubiquitous [7

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) 405, 546–550 (2000). [CrossRef] [PubMed]

]. To date, there is solely one landmark experiment demonstrating both Zeno and anti-Zeno effects in a truly unstable quantum system [15

15. M.C. Fischer, B. Gutierrez-Medina, and M.G. Raizen, “Observation of the Quantum Zeno and Anti-Zeno Effects in an unstable system,” Phys. Rev. Lett.87, 040402-1-040402-4 (2001). [CrossRef] [PubMed]

].

Exploitation of quantum-classical analogies, on the other hand, has been used on many occasions to mimic and visualize at a macroscopic classical level some basic quantum dynamical features characteristic of the Schrödinger equation that have difficult or even impossible access in quantum systems (see e.g. [16

16. See, for instance: D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, Berlin, 2004) and references therein.

]). In particular, it was recently theoretically demonstrated [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

, 18

18. S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

] that tunneling escape of light waves in an optical waveguide side-coupled to a semi-infinite array (from now on referred to as ‘semi-array’) exactly mimics the decay dynamics of a truly unstable quantum state coupled to a continuum, thus providing a classical realization (‘optical Zeno effect’) of the non-exponential decay behavior in the Friedrichs-Lee model commonly used in theoretical studies of the quantum Zeno effect [6

6. M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A 61, 022105-1-022105-5 (2000). [CrossRef]

, 7

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) 405, 546–550 (2000). [CrossRef] [PubMed]

, 8

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

, 9

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

, 19

19. I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, “Quantum Zeno and anti-Zeno effects in the Friedrichs model,” Phys. Rev. A 63, 062110-1-062110-10 (2001). [CrossRef]

]. In an optical system, projective measurements (corresponding to the collapse of the wave function according to von Neumann’s projection postulate) are simply mimicked by sequential spatial alternations of the semi-array. Each interruption restarts the decay process erasing the memory of the continuum and therefore ideally corresponds to a “wave collapse” in the quantum realm [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

]. The spatial length between successive interruptions plays the role of the time interval between successive quantum measurements. The use of an engineered photonic structure to mimic at a classical level the Zeno dynamics, far from being a mere curiosity, offers the rather unique advantage of a direct and precise mapping of the space-time evolution of the Schrödinger probability density which is not fully accessible for quantum tunneling.

We have recently shown [20

20. G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. 90, 261118-1-261118-3 (2007). [CrossRef]

] that scanning tunneling optical microscopy (STOM) [21

21. S.I. Bozhevolnyi and L. Kuipers, “Near-field characterization of photonic crystal waveguides,” Semicond. Sci. Technol. 21, R1–R16 (2006). [CrossRef]

] based on a hollow pyramid mounted on a silicon cantilever provides an excellent tool for accurate quantitative tracing of the flow of discretized light in evanescently-coupled optical waveguides [22

22. D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

], allowing for high reproducibility and easy interpretation of the data. In this work we report on the first experimental demonstration, by using the same STOM setup, of the optical Zeno effect for the decay of light waves in an optical waveguide.

2. Sample and methods

The structures designed for the experimental demonstration of the optical Zeno effect are shown in Fig. 1 and consist of a single-mode straight channel waveguide (W) which is side-coupled either to a semi-array (S) or to a set of alternating semi-arrays (S1, S2, etc.) of 16 identical waveguides in the geometries shown in Figs. 1(a) and 1(b), respectively.

Fig. 1. (a) Sketch of the sample with the central waveguide W coupled to the semi-array S. Δ0 is the coupling rate between Wand S, Δ the coupling rate between adjacent waveguides in S, a 0 and a are waveguide distances. (b) Sketch of the sample with the central waveguide W and the alternating semi-arrays S1, S2, etc. for the demonstration of the optical Zeno effect. τ is the distance between successive interruptions of the semi-arrays. (c) Topography map, obtained by means of atomic force microscopy, of the sample shown in panel (b). (d) Sketch of the energy levels for the quantum analogue describing a discrete level |χ〉 coupled to a tight-binding continuum |ω〉 of width 4ħΔ.

3. Model

Light propagation along the paraxial direction z of a weakly waveguiding structure is described by a Schrödinger-like equation for the electric field amplitude Ψ(x, y, z), which exactly mimics the temporal evolution of a quantum particle in a two-dimensional multiple-well potential V(x, y)≃ns-n(x, y), where n(x, y) is the refractive index profile of the waveguide-array system and ns the substrate refractive index. By substituting ħλ/2π, mns (m being the mass of the particle in the Schrödinger equation), and tz, the temporal evolution of the particle wave function in the quantum problem is replaced, in our photonic structure, by the spatial propagation of the electric field amplitude Ψ(x, y, z) along z [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

]. If the system is initially prepared with the particle in the well W, it can evolve via tunneling into the chain of adjacent wells and the probability P to find the particle in the well W decays with time toward zero in absence of bound surface states [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

]. The decay of a discrete level |χ〉 into a tight-binding continuum |ω〉 of states is commonly described in terms of the Friedrichs-Lee Hamiltonian H = H 0+HI (see e.g. [7

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) 405, 546–550 (2000). [CrossRef] [PubMed]

, 8

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

, 9

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

]), where

H0=h¯σχχ+2Δ2Δdωh¯ωωω
(1)

is the free Hamiltonian,

HI=h¯dω[g(ω)ωχ+h.c.]
(2)

is the interaction Hamiltonian, g(ω) is the discrete-continuum spectral coupling amplitude, 4 ħΔ is the width of the continuum band, and σ accounts for a possible detuning between the position of the discrete level and the center of the continuum, as shown in Fig. 1(d). For the system of our interest, the spectral coupling amplitude takes the specific form [18

18. S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

]

g(ω)=Δ0πΔ[1(ω2Δ)2]14
(3)

where Δ0 measures the strength of the discrete-continuum coupling.

cχ(z)=12π0+i0++idsexp(sz)isσ(s),
(4)

where

(s)=dωg(ω)2isω=iΔ022Δ2[ss2+4Δ2]
(5)
Fig. 2. Decay behavior of the fractional power trapped in the waveguide W for the sample in Fig. 1(a), as given by Eq. (4) (solid line) and by a full beam propagation analysis (dashed line). In the inset, the effective decay rate γeff(z)=-(1/z)lnP(z) (solid line) is compared to the ‘natural’ decay rate γ 0 (dotted line).

is the self-energy. For |σ|<2-(Δ0/Δ)2, there are no bound states and cχ (z) (and hence the survival probability P(z) = |cχ(z)|2) asymptotically decays to zero. However, the decay law shows non-exponential features which can be markedly pronounced [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

].

For the waveguide structure manufactured for our experiment, the boundary waveguide W and the waveguides in the semi-arrays have approximately the same refractive index profile, i.e. we may assume σ/Δ≃0. For the waveguide separations of the fabricated structure (a = 9.5 µm and a 0 = 11 µm), the values of the coupling rates between the waveguides turn out to be Δ≃0.435 mm−1 and Δ0≃0.223 mm−1, as measured using two reference directional couplers (see e.g. [25

25. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express 15, 1579–1587 (2007). [CrossRef] [PubMed]

]). Solid line in Fig. 2 shows, correspondingly, the predicted decay law for P(z) in the waveguide W of Fig. 1(a), as obtained by Eq. (4). The inset in Fig. 2 shows the corresponding behavior of the effective decay rate γ eff(z)=-(1/z)lnP(z), compared to the ‘natural’ constant decay rate γ0=2(Δ0Δ)21(Δ0Δ)2 , obtained in the frame of Gamow’s approach to quantum tunneling decay [9

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

, 17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

], which would give an exponential decay for P(z). The spatial length τ = 3.1mm between successive interruptions in the sample of Fig. 1(b) plays the role of the time interval between successive measurements, and for the observation of the Zeno effect it should be chosen smaller than a characteristic ‘Zeno time’ τ Z, where τ Z is the smallest root of the equation γ eff(τ Z)=γ 0 [9

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

]. In our case, one can estimate τ Z ≃ 30 mm, i.e. much longer than the sample length (~12 mm), and anti-Zeno effect can be a priori excluded. Dashed line in Fig. 2 shows the same decay behavior simulated by means of a fully numerical integration of the paraxial wave equation for Ψ(x, y, z) using a beam-propagation software [26

26. BeamPROP, 5.0 ed., Rsoft Design Group, Inc., 2002.

]. P(z) is calculated by projecting, at each propagation distance z, the envelope Ψ(x, y, z) over the fundamental mode of waveguide W. The good agreement between the two simulations further supports the use of a tight-binding approach (coupled-mode equations) to tackle the optical problem.

4. Results and discussion

The detailed experimental spatial maps for the light intensity distribution |Ψ(x,0, z)|2 in the (x, z) plane of the sample are shown in Figs. 3(a) and 3(b) for the samples of Figs. 1(a) and 1(b), respectively.

The maps are obtained by taking successive STOM scans along the propagation z direction with steps of Δz = 500µm. Each scan covers a sample area of size 170µm×20µm in the (x, z) plane, thus comprising the main waveguideW and all the 16 waveguides in the semi-arrays S, S1, S2, etc. From the acquired STOM images, the integrated optical signal along the x direction is normalized in order to take absorption and internal losses into account. In this way the decay law for the light power trapped in waveguide W can be calculated, as fully described in [20

20. G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. 90, 261118-1-261118-3 (2007). [CrossRef]

]. In Fig. 3(c) we show the experimental decay curves for the samples of Figs. 3(a) (squares) and 3(b) (circles). The difference in the decay behavior between the two curves clearly shows that, as anticipated before, slow down of the decay due to frequent projective measurements is obtained when the light trapped in waveguideWis alternately coupled to the right and left semi-arrays S1, S2, etc. (optical Zeno effect). Each interruption of the semi-arrays restarts the decay process in the waveguide W and thus mimics a projective (von Neumann) measurement in the corresponding quantum mechanical problem [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

]. The spatial distance τ between successive interruptions is large enough to ensure that light trapped in the semi-array after each interruption is scattered out into the substrate in a short distance (~600 µm). In this way, the ‘memory’ in the continuum is erased after each interruption, as shown in the inset of Fig. 3(c), where the normalized power decay law in each of the semi-arrays is plotted and superimposed to the theoretical prediction.

Fig. 3. Experimental spatial maps, acquired by STOM imaging, for the normalized light intensity distribution in the waveguides of the samples described in Figs. 1(a) (panel a) and 1(b) (panel b). Panel c shows the experimental decay curves for the fractional power trapped in the waveguide Wfor the sample of panel a (squares) and panel b (circles). Solid lines represent the numerical prediction based on Eq. (4). In the inset, normalized power decay laws are plotted for each semi-array for the sample in panel b.

5. Conclusion

In conclusion, we reported on the experimental demonstration of a classical analogue of the quantum Zeno effect for light waves [17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

] in an engineered photonic structure. Escape of light waves via optical tunneling in a waveguide side-coupled to a semi-array has been accurately measured by STOM imaging using a cantilevered hollow pyramid tip, showing that periodic interruptions of the decay result in a deceleration of the decay process.

We gratefully acknowledge A. Bassi, G. Cerullo, V. Foglietti, M. Lobino, M. Marangoni, and M. Savoini for fruitful discussions and for their support during the experimental sessions.

References and links

1.

B. Misra and E. C. G. Sudarshan, “The Zeno’s Paradox in Quantum Theory,” J. Math. Phys. 18, 756–763 (1977). [CrossRef]

2.

W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A 41, 2295–2300 (1990). [CrossRef] [PubMed]

3.

P. Knight, “Watching a Laser Hot-Pot,” Nature (London) 344, 493–494 (1990). [CrossRef]

4.

H. Nakazato, M. Namiki, and S. Pascazio, “Temporal behavior of Quantum Mechanical Systems,” Int. J. Mod. Phys. B 10, 247–295 (1996). [CrossRef]

5.

A.G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A 54, R3750–R3753 (1996). [CrossRef] [PubMed]

6.

M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A 61, 022105-1-022105-5 (2000). [CrossRef]

7.

A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) 405, 546–550 (2000). [CrossRef] [PubMed]

8.

A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

9.

P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. 86, 2699–2703 (2001). [CrossRef] [PubMed]

10.

P. Facchi and S. Pascazio, Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2001), Vol. 42, p. 147.

11.

E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. 97, 260402-1-260402-4 (2006). [CrossRef]

12.

P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. 89, 080401-1-080401-4 (2002). [CrossRef] [PubMed]

13.

K. Koshino and A. Shimizu, “Quantum Zeno Effect for exponentially decaying systems ,” Phys. Rev. Lett. 92, 030401-1-030401-4 (2004).

14.

A. Peres, “Zeno Paradox in Quantum Theory,” Am J. Phys.48, 931–932 (1980); G.S. Agarwal and S. P. Tewari, “An all-optical realization of quantum zeno effect,” Phys. Lett. A 185, 139–142 (1994); M. Kitano, “Quantum Zeno Effect and intracavity polarization filters,” Opt. Commun. 141, 39–42 (1997); V. Kidambi, A. Widom, C. Lerner, and Y. N. Srivastava, “Photon polarization measurements without the Quantum Zeno Effect,” Am. J. Phys. 68, 475–481 (2000); K. Yamane, M. Ito, and M. Kitano, “Quantum Zeno Effect in Optical Fibers,” Opt. Commun. 192, 299–307 (2001). [CrossRef]

15.

M.C. Fischer, B. Gutierrez-Medina, and M.G. Raizen, “Observation of the Quantum Zeno and Anti-Zeno Effects in an unstable system,” Phys. Rev. Lett.87, 040402-1-040402-4 (2001). [CrossRef] [PubMed]

16.

See, for instance: D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, Berlin, 2004) and references therein.

17.

S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

18.

S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

19.

I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, “Quantum Zeno and anti-Zeno effects in the Friedrichs model,” Phys. Rev. A 63, 062110-1-062110-10 (2001). [CrossRef]

20.

G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. 90, 261118-1-261118-3 (2007). [CrossRef]

21.

S.I. Bozhevolnyi and L. Kuipers, “Near-field characterization of photonic crystal waveguides,” Semicond. Sci. Technol. 21, R1–R16 (2006). [CrossRef]

22.

D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] [PubMed]

23.

G. Della Valle, S. Taccheo, P. Laporta, G. Sorbello, E. Cianci, and V. Foglietti, “Compact high gain erbium-ytterbium doped waveguide amplifier fabricated by Ag-Na ion exchange,” Electron. Lett. 42, 632–633 (2006). [CrossRef]

24.

AlphaSNOM, WITec GmbH, Ulm, Germany.

25.

A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express 15, 1579–1587 (2007). [CrossRef] [PubMed]

26.

BeamPROP, 5.0 ed., Rsoft Design Group, Inc., 2002.

OCIS Codes
(000.1600) General : Classical and quantum physics
(080.1238) Geometric optics : Array waveguide devices
(180.4243) Microscopy : Near-field microscopy
(070.7345) Fourier optics and signal processing : Wave propagation

ToC Category:
Microscopy

History
Original Manuscript: February 1, 2008
Revised Manuscript: March 1, 2008
Manuscript Accepted: March 2, 2008
Published: March 6, 2008

Citation
P. Biagioni, G. Della Valle, M. Ornigotti, M. Finazzi, L. Duò, P. Laporta, and S. Longhi, "Experimental demonstration of the optical Zeno effect by scanning tunneling optical microscopy," Opt. Express 16, 3762-3767 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3762


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References

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  20. G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duo, and M. Finazzi, "Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy," Appl. Phys. Lett. 90, 261118-1-261118-3 (2007). [CrossRef]
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  26. BeamPROP, 5.0 ed., Rsoft Design Group, Inc., 2002.

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